Inspired by this question Symplectic structure of $TS^{n-1}$ we ask:
What is an example of a manifold $M$ whose cotangent bundle $T^*M$ is an Stein manifold but the canonical symplectic structure of $T^*M$ is equivalent to no symplectic structure imposed on $T^*M$ via a holomorphic embedding in some $\mathbb{C}^n$? In the other word: we search for a manifold $M$ such that $T^*M$ is a holomorphic submanifold of some $\mathbb{C}^n$ so it admits some symplectic structures inherited frm $\mathbb{C}^n$ but non of this structures obtained in this way is symplectomorphism to the canonical structure of the cotangent bundle $T^* M$.
Note; It seems that the assumption "The cotangent bundle is an Stein manifold" is redundant". Namely it seems that every cotangent bundle is an Stein manifold. Since The whole cotangent bundle is diffeomorphic to the disc bundle $T^*_{r}M$, the disc bundle of radius $r$. In the following conversation one observe that there exist an Stein structure on the disc bundle. Please see the link bellow: How does one complexify a real $n$-dimensional Riemannian manifold $(M,g)$?
A Confession: I confess I did not read the details of the paper of Bruhat-Whitney and the paper of Lempert. But I guess that their constructions leads us to a negative answer to this post.
